J. Phys. Chem. C 2008, 112, 16991–16998
16991
Quantum Interference: The Structural Dependence of Electron Transmission through
Model Systems and Cross-Conjugated Molecules
David Q. Andrews,* Gemma C. Solomon,* Randall H. Goldsmith, Thorsten Hansen,
Michael R. Wasielewski, Richard P. Van Duyne, and Mark A. Ratner
Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208
ReceiVed: June 24, 2008; ReVised Manuscript ReceiVed: August 25, 2008
We report on a class of molecules that exhibit nonlinear current/voltage behavior in the low bias tunneling
regime. This interesting behavior is attributed to quantum interference. Using site models, we show that
interference features, while common, do not necessarily occur at experimentally relevant energies, hindering
realization in transport measurements. Calculations made using a nonequilibrium Green’s function code show
that quantum interference can be experimentally relevant in cross-conjugated molecules. A detailed bond
length analysis of cross-conjugated molecules gives insight into why these molecules have interference at
energetically accessible regions. The interference features are shown to be stable to both an electronic dephasing
analysis and geometric fluctuations provided by molecular dynamics.
Introduction
The potential for molecular electronics is rooted in the unique
chemical properties of molecules. The suggestion that single
molecules could be used as discrete electronic components was
made over 30 years ago.1 Since then, experimental studies of
charge transport through molecules have largely been completed
in an electrochemical transport junction or optical donorbridge-acceptor context. These measurements have been
completed on a large range of molecular systems as well as
cross-conjugated molecules. In junction measurements, the
molecular conductance is studied, while in optical measurements
the transport through the molecules is measured as a charge
separation or recombination rate. Single molecule two probe
transport measurements have opened up a new area of research
where the average transmission through a molecule can be
studied without charging the molecule, allowing the ability to
tune the incident electron energy.2-4 If the charge transport
behavior of a molecule is extremely sensitive to incident electron
energy near the Fermi level, this control of voltage bias should
yield new information on the transport behavior of molecules.
Measuring transport through single molecules has largely
focused on molecular wires. These generally consist of linear
molecules (both saturated and unsaturated) or fully conjugated
aryl systems. Making such measurements is critical in establishing new experimental techniques and allows comparison among
different methods. There has also been strong theoretical
development of models to explain the transport behavior in a
metal-molecule-metal junction.5-9 This field of research is
relatively young, and the transport through less common organic
structures should open up new areas where the electron
transmission as a function of energy behaves in unique ways.
In analyzing the charge transport through molecules in the
coherent tunneling limit, the Simmons equation, which pictures
a molecular junction as a tunneling barrier where the height of
the barrier is essentially the distance from the Fermi level to
the nearest molecular resonance, is often used.10-17 This picture
incorporates a common illustration showing the discrete mo* Corresponding author. E-mail: [email protected] (D.Q.A.),
[email protected] (G.C.S.).
Figure 1. The transmission probability is a sum of the couplings
between source/donor-molecular orbitals-drain/acceptor for a single
channel. The electron “sees” all available molecular orbital pathways
in energy space. This can be compared to Young’s double slit
experiment; in electron transport, destructive interference can occur
when summing the coupling terms across the system.18
lecular frontier energy states sandwiching the electrode Fermi
level. Within this picture, explaining conductance relies on four
main factors, the electrode/molecule coupling strength, the
associated molecular state broadening, the alignment and
distance of the Fermi energy to the frontier molecular orbitals,
and the energy level shift of the molecular states upon contact
to the electrode.
While this picture has proven extremely useful, it is based
on an independent electron model that assumes the molecular
energy states do not interact in the transport process. We show
here that interactions between molecular orbital energy states,
sometimes including those far from the Fermi level, can have
a very large effect on the transport properties of a molecule in
the tunneling regime at low bias voltage. Figure 1 shows a
diagram illustrating schematically how the transmission probability for an electron to tunnel through a molecule is calculated
as the sum of the coupling terms from donor to acceptor.18 The
coupling terms between donor and acceptor vary with the
geometry, orientation, and electrode linking sites of the molecule, and a single molecule can have drastically different
conductance behavior for different linking geometries even
though the frontier molecular orbitals are similar.
The largest deviation from the Simmons model of charge
transport occurs when the coupling terms across the molecule
10.1021/jp805588m CCC: $40.75  2008 American Chemical Society
Published on Web 10/07/2008
16992 J. Phys. Chem. C, Vol. 112, No. 43, 2008
destructively interfere in the energy region between the highest
occupied molecular orbital (HOMO) and the lowest unoccupied
molecular orbital (LUMO). In the context of electron transport,
interference can occur among the terms that are summed to
calculate the transmission probability. At the energy level of a
molecular resonance, the transmission probability f 1; conversely, when there is quantum interference, an antiresonance
can occur where the transmission probability f 0.
Interference features have been studied extensively in quantum dots.19-22 Quantum dots provide a discrete energy spectrum
similar to molecules and, due to the increased size scale, enable
greater experimental control. In the quantum dot literature,
measurements have been made on the transmission phase in a
quantum dot using interferometers.19,23 In this work, the
transmission phase change going through a resonance was π,
but unexpected were the measured abrupt phase changes of π
between sequential resonances.23 This abrupt phase change was
later attributed to a nonspanning node in the specific energy
level of the quantum dot.24 A nonspanning node is defined as
an in-phase resonance where the change in wave function
density leads to a node that touches only one boundary or zero
boundaries of a quantum dot.24 In the limit where a quantum
dot can be treated as a large molecule, we show how the phase
change in molecular resonances directly correlates to the
nonspanning node explanation.
In visual similarity to cross-conjugated molecules, T-shaped
quantum dots have been proposed as quantum interference
devices.22,25 Experimentally observing interference effects in
nanostructured materials is hindered by the need to maintain
electron phase coherence over long distances.20,21 Here, we
discuss systems that are similar to a T-shaped quantum dot
system but do not require extremely long electron coherence
lengths; specifically, we consider cross-conjugated molecules.
In focusing our research on single molecule effects, it is
important to realize that experimental measurements of single
molecule junctions are quite tricky. Controlling the formation
of an angstrom-nanometer scale gap as well as maintaining
contact to both electrodes is extremely difficult. Once these
measurements are made, care has to be taken in analyzing the
data. It has become apparent through many calculations and
experimental measurements that molecules can sample many
different geometries and possible mechanistic regimes with
respect to time. These fluctuations in current can be of the same
order of magnitude as the observable.26
Because of the difficulties in controlling small molecular
junctions, many early measurements have been reanalyzed.27-29
For useful device applications, a molecule with novel transport
behavior should be stable to geometric fluctuations while still
providing interesting transport properties that can be experimentally realized.
To investigate interference features, we ask the questions:
How can interference occur in model systems? Why is the
interference present in cross-conjugated molecules? How stable
is this interference feature to electronic dephasing and molecular
vibrations?
Two- and four-site model systems are studied to give an
indication of how interference features can occur in a transport
picture and how interference features appear to be quite
common. A detailed analysis of cross-conjugated bonds is
presented to give an understanding of why this molecular unit
has such a large effect on electron transport. Using nonequilibrium Green’s function codes, we calculated the transmission
and current-voltage characteristics for a series of crossconjugated molecules. Using molecular dynamics and electronic
Andrews et al.
Figure 2. Transmission through a two-site model. The electrode
attachments determine the presence of an antiresonance feature. The
plot on the left shows transmission as a function of energy, and the plot
on the right shows phase as a function of energy. The antiresonance
feature in the perpendicular model is seen at E ) 0 as a zero in
transmission and a corresponding phase jump of π.
dephasing calculations, we show that the interference feature
calculated in cross-conjugated molecules is stable.
A Two-Site Model
To understand how quantum interference can occur in electron
transport through molecules, it is illustrative first to analyze a
donor-bridge-acceptor model of a two-site system. As shown
in Figure 2, there are two possible electrode/molecule site
linkages (hereafter simply referred to as binding site geometries)
for a two-site bridge between a donor and an acceptor. We refer
to these two binding site geometries as the linear model and
the perpendicular model. In the linear model shown in Figure
2a, the donor is connected to one site and the acceptor to the
other site. In the perpendicular model shown in Figure 2b, the
donor and acceptor are both connected to the same site. Setting
the site energy to 0 and the intersite coupling, β, to -0.5, we
calculate the Hückel molecular orbitals, for the isolated twosite system as shown.
The energy-dependent electronic coupling between the electrodes determines the transport properties of a molecular
junction. To calculate the coupling between the donor and the
acceptor, we solve the one-electron Hückel Hamiltonian. This
Hückel coupling matrix is then transformed into a molecular
orbital basis.
The transmission is calculated using the Landauer equation:30,31
T(E) ) Tr[ΓL(E)Gr(E)ΓR(E)Ga(E)]
ΓL/R
(1)
Gr/a
with
as the left and right spectral densities and
as the
retarded and advanced Green’s functions. Using a transformation
to a molecular conductance orbital basis,32 we have shown that
the transmission can be calculated using the sum of the squared
couplings tRβ through each molecular orbital (labeled by i),18
given by
tRβ(E) )
R
L †
r
(E)
(E)Gii
(E)Viβ
∑ VRi
(2)
i
where R and β are the basis dimensions of the electrode, VL(R)
are the bridge-electrode couplings, and Gr′ is the retarded
Green’s function18 (calculation of the transmission from the
coupling is shown in the Supporting Information).
To calculate the coupling as a function of energy, we leave
the site energies at 0 and sweep the electrode energy. In both the
linear model and the perpendicular model, the coupling through
Quantum Interference
J. Phys. Chem. C, Vol. 112, No. 43, 2008 16993
Figure 3. Four-site model system with six possible donor-acceptor couplings. In all model systems with a side group there is an antiresonance
feature, although not necessarily at E ) 0.
the highest occupied molecular orbital (HOMO) level is
identical. The coupling through the lowest unoccupied molecular
orbital (LUMO) differs only in a sign change. The overall
coupling through the system can be determined by taking the
absolute value of the sum of the individual coupling terms.
At E ) 0, the sum of the coupling terms in the perpendicular
two-site model is zero. The coupling terms through the HOMO
and LUMO molecular orbitals are equal and opposite, leading
to destructive interference. This destructive quantum interference
gives the perpendicular two-site model the sharp antiresonance
at E ) 0. To compare directly with two probe measurements,
we calculate the transmission as a function of energy18 as shown
in Figure 2c.
In Figure 2d, we show the phase of the transmission as a
function of energy. The transmission phase is defined as the
arctangent of the imaginary/real parts of the transmission.18 As
the energy passes a molecular resonance, the transmission phase
always changes by π. In the perpendicular two-site model, there
is an abrupt phase change of -π at E ) 0, the location of the
antiresonance. This antiresonance occurs whenever there is a
zero in both the real and the imaginary plane of the transmission.
A zero in both the imaginary and the real components of the
transmission need not cause a phase jump of π, and for this
reason we have included the more complex parametric plots
(shown in the Supporting Information). Sweeping the energy
from very low to very high, the transmission phase changes by
nπ, where n is equal to the smallest number of sites between
the donor and acceptor or electrode binding sites. In the
perpendicular two-site model, there is only one site between
donor and acceptor, indicating that there will be one abrupt phase
change of -π.
Using a two-site model, we can see that an antiresonance
occurs at E ) 0 for the perpendicular model. This zero in the
transmission is a direct result of the destructive interference
caused by the coupling to the HOMO and the LUMO orbitals.
In this two-site model, the antiresonance is correlated with either
no change or a decrease in the number of spanning nodes
between the donor and the acceptor. A spanning node is taken
here to be a node in the shortest path between the donor and
acceptor.
A Four-Site Model
We now look at the coupling through a four-site model shown
in Figure 3. The Hückel molecular orbitals for the isolated foursite system are shown in the upper left. As with the two-site
model, the isolated molecular orbitals are the same, independent
of how we connect the donor and acceptor. In the four-site
model, there are 10 different ways to connect the donor and
the acceptor. We limit our discussion to the transmission plots
for six selected systems. Figure 3b compares the transmission
through D1A4 and D1A2. There are two antiresonances, both
occurring in the D1A2 model. The antiresonance occurs when
there is a molecular orbital that introduces a nonspanning node,
taken here to be a node that is not in the shortest path between
the donor and acceptor.
In Figure 3c, D1A3 and D2A3 both have an antiresonance
at E ) 0 in the transmission. In both models, a nonspanning
node is introduced in the LUMO. The HOMO (E ) 0.618β) in
Figure 3a has one node in the electron density between site
pairs 1 and 3, and 2 and 3. The LUMO (E ) -0.618β) has
only 1 node between sites 1 and 3 and 0 nodes between sites 2
and 3. In the D2A3 model, there are two nonspanning nodes
introduced, causing the width of the antiresonance to increase.
Figure 3d shows two models, D1A1 and D2A2, where the
donor and acceptor are both connected to the same site. In this
example, each subsequent molecular orbital introduces a nonspanning node in the Hückel orbitals. Correspondingly, in the
transmission plot for both of these four-site models, there is
antiresonance between each subsequent higher energy molecular
16994 J. Phys. Chem. C, Vol. 112, No. 43, 2008
Figure 4. Resonance structures for a cross-conjugated molecule. The
electron delocalization shown in (b) and (d) illustrates how in a threeway junction the electron can only delocalize in two directions; there
is no direct electron delocalization possible between atoms 1 and 5.
resonance. The antiresonances occur at the energies of the
molecular orbitals of the isolated side chain.
The two- and four-site models show that quantum interference
and corresponding antiresonances are quite common in toy
systems. Within these controlled systems, an antiresonance
occurs whenever there is a zero in both the real and the
imaginary portion of the transmission. These transmission zeros
appear whenever there is a molecule having sites not in the
shortest geometric path between donor and acceptor. In the
molecular orbitals, this zero is seen as the introduction of a node
in the molecular orbitals that does not span the path between
donor and acceptor. While this feature may be common in toy
systems, it has been noticeably absent in experimental measurements. This absence is due mainly to the occurrence of
interference features not energetically positioned between the
HOMO and LUMO.33 If the interference is not near the Fermi
energy, experimental verification is difficult and the appearance
of the antiresonance may have little effect on charge transport.
One commonly studied molecule that has an interference feature
near the Fermi energy is benzene connected to the electrodes
through the meta positions. Experimental verification of these
differences has been noted in I/V measurements in a mechanical
break junction,34 in absorbance measurements,35 as well as in
numerous theoretical calculations.35-41 We will now focus on
the acyclic cross-conjugated bond, a common molecular bonding
motif that has been calculated to have interference between the
HOMO and LUMO.42
Cross-Conjugated Molecules
“A cross-conjugated compound may be defined as a compound possessing three unsaturated groups, two which although
conjugated to a third unsaturated center are not conjugated to
each other. The word “conjugated” is defined here in the
classical sense of denoting a system of alternating single and
double bonds.”43 By definition and as illustrated in the resonance
picture shown in Figure 4, cross-conjugation in transport
junctions depends on the position linking to the electrodes,
reminiscent of our observations with the toy models. In a
molecule that is fully π-conjugated, this binding dependence
will affect the low bias delocalization when there are an odd
number of carbon atoms between the donor and acceptor.
Electron delocalization and electron transport are correlated,44,45
and from Figure 4 it seems evident that electron transport should
behave differently if the electrodes are connected to atoms 1
and 6 versus connection to atoms 1 and 5, as there is no
delocalization between atoms 1 and 5.
Andrews et al.
Cross-conjugation is a common motif in organic chemistry,
found in a wide range of naturally occurring and synthetic
molecules; yet detailed analysis of the electron delocalization
and transport properties is sparse. Recent synthetic work has
detailed a large number of cross-conjugated molecules.46,47 A
number of theoretical studies of HOMO’s and LUMO’s of crossconjugated molecules have shown that the electron delocalization is split at the cross-conjugated unit.44,48 This difference
between linear and cross-conjugation has been borne out in
absorbance measurements, as the cross-conjugated unit results
in a measured blue shift in the absorbance peak.49 In
donor-bridge-acceptor systems, the cross-conjugated connectivity leads to measured electron transfer rates lower than
those of linearly conjugated systems but generally higher than
those of nonconjugated systems.50
Bond Length Analysis
The molecules shown in Figure 5 are used to illustrate the
fundamental bond length differences between cross conjugation,
linear conjugation, and saturated structures. It has been previously established that bond length and bond length alternation
are correlated to π electron delocalization in both linear and
cross conjugation molecules.44,45,51,52 In this analysis, we
compare the bond lengths for 15 molecules with three different
central bonding motifs; cross conjugation, linear conjugation,
and a saturated carbon atom. To allow direct comparison with
metal-molecule-metal transport calculations presented later,
all of the molecules are thiol terminated, and the geometry
optimized gas-phase structures are calculated using density
functional theory with B3LYP53,54 and 6-311G** in Q-Chem
3.0.55 Our analysis focuses on the delocalization of the charge
between the sulfur atoms because this will be the direction of
charge transport in a molecular junction. The cross-conjugated
molecules labeled 6-10 all have a double bond perpendicular
to the direction of charge transport. Within this subset of
molecules, 6, 7, and 9 meet the specific definition of cross
conjugation. Molecules 11-15 have a double bond connected
in the trans orientation between the thiol terminations. Molecules
11, 12, and 14 are completely linearly conjugated between thiol
groups. Molecules 1-5 are the same as molecules 6-10, but
with hydrogen atoms replacing the cross-conjugated double
bond.
To simplify the comparison among so many molecules, we
focus on the bonds labeled a, b, and c in Figure 5. Bonds a and
c are both formally single bonds, and bond b is a double bond.
The carbon single bonds are systematically longest in the
molecules 1-5 and shortest in the molecules with linear
conjugation, 11-15. The corresponding bond lengths in the
cross-conjugated molecules 6-10 are between the length of the
saturated and unsaturated linear molecules. The changes in bond
lengths agree well with previous studies completed on molecules
1, 2, 6, 7, and 9.45
To compare the carbon single bond lengths, we have
calculated the percentage of double bond character. This is done
by comparing all of the bond lengths with those calculated for
ethene and ethane. Ethene by definition has 100% double bond
character, and ethane has 0% double bond character. This bond
length comparison shows that in a cross-conjugated molecule
the π electron delocalization is less than that in a linearly
conjugated molecule but greater than that in a fully saturated
carbon atom.
Using the bond length characterization to analyze these data,
we come to three important conclusions. First, a crossconjugated molecule and a linearly conjugated molecule show
Quantum Interference
J. Phys. Chem. C, Vol. 112, No. 43, 2008 16995
Figure 5. A bond length analysis for a series of molecules with sections of conjugated, nonconjugated, and cross-conjugated carbon atoms. Red
indicates bond a or c is bonded to an alkyne group, blue to an alkene group, and green to an alkane group. By definition, the ethane bond length
has 0% double bond character, and the ethene bond length has 100% double bond character. A negative double bond character indicates a
carbon-carbon bond longer than that in ethane.
Figure 6. Transmission and current-voltage calculated in Hückel IV 3.0 for molecules 4, 5, 9, and 14. In the transmission plot shown on the left,
the cross-conjugated molecule 9 shows an interference feature at E ) 0. The corresponding current-voltage plot on the right shows that current
through molecule 9 has a greater voltage dependence than molecules 4, 5, and 14 (the deviation from linearity in molecule 9 is discussed in the
Supporting Information).
different behavior with respect to electron delocalization.
Second, in cross-conjugated molecules, the electron delocalization is reduced between cross-conjugated regions of the
molecules, as is evident by the increase in both C-C bond
lengths. The saturated C-C bond length only changes due to
direct coupling to an unsaturated carbon. This effect is local,
and the bond length changes are largely a nearest neighbor
effect. Within the three sets of molecules, 1-5, 6-10, and
11-15, the C-C bond lengths a and c remain weakly correlated
to each other (except when required to match by symmetry in
molecules 1, 4, 6, 9, 11, and 14), implying that the addition of
the cross-conjugated bond induces very little electron delocalization across the cross-conjugated carbon atom.
Figure 7. Electronic dephasing effects calculated for meta- and parabenzene as compared to linear and cross-conjugated site representations.
The decay time is defined as the time necessary for the total population
to decrease 95%.
Transport Calculations
To calculate the transport properties of cross-conjugated
molecules, we look at a representative series of molecules from
Figure 5, specifically molecules 4, 5, 9, and 14. These molecules
represent a complete series of double bonded molecules with
cross-conjugation, linear conjugation, and replacing the crossconjugated unit with a saturated carbon atom or saturation of
one double bond. The transport calculations are completed in
the Laundeur-Imry low bias tunneling regime.30,31,56,57 Figure
6 shows a series of transmission spectra and I/V plots for this
family of molecules. These calculations were completed in
Hückel IV 3.0.7,58 The qualitatively similar plots calculated using
ATK 2.0.459-62 are shown in the Supporting Information.
The transmission plot for the cross-conjugated molecule, 9,
has HOMO and LUMO molecular orbitals at an energy and
16996 J. Phys. Chem. C, Vol. 112, No. 43, 2008
Andrews et al.
Figure 8. Molecular dynamics results for molecules 5, 9, and 14. Plots (a) and (b) show 100 transmission traces calculated for molecules 9 and
14. Plots (c) and (d) show Gaussian fits to the distribution of conductance values calculated at 10 mV and 2 V.
with transmission similar to those of the fully linearly conjugated
molecular 14. Yet at E ) 0 the transmission through molecule
9 is over 3 orders of magnitude lower. The transmission through
the cross-conjugated molecule 9 is also lower than that through
molecules 4 and 5. This remarkable result occurs because of
quantum interference: the coupling terms in the cross-conjugated
unit cancel the π contribution to transport near the Fermi energy
almost completely. For an antiresonance to occur, a zero in the
transmission is required. In this example, there is complete
destructive interference in the dominant π transport channel,
but there is still transport through the σ channel. In the I/V plot
shown on the right in Figure 6, the current through molecule 9
is the lowest at low voltage but increases at a higher nonlinear
rate than any of the other molecules. The molecules 4, 5, and
14 behave more like resistors with approximately linear voltage
dependence (shown in the Supporting Information).
Electronic Dephasing
Interference stability is of primary importance for experimental measurement of the unique properties of cross-conjugated
molecules. Electronic dephasing,63-65 frequently caused by
fluctuations in molecular environment or geometry, might lead
to decoherence of the transmitted electron and destruction of
the interference feature.66,67 In earlier studies, we looked at how
the traditional reactivity series in ortho-, meta-, and parabenzene could be recast as an interference effect67 and how this
effect could be erased by purely local dephasing.66,67 The
calculation of transport dynamics is done using the quantum
Liouville equation with dephasing included by reducing the
magnitude of the off-diagonal elements of the density matrix
(coherence).68,69 At t ) 0, all population is placed on the donor/
source site, while an absorbing boundary condition on the
acceptor/drain site is used to simulate irreversible electron
transfer. We have calculated the decay time necessary for the
total system population to decrease to 5% of the initial value,
shown in Figure 7. Our results indicate that a dephasing strength
of γ > 100 cm-1 is necessary for coalescence of the transport
time through both two-site models.
Measuring or estimating an absolute electronic dephasing
strength can be difficult with estimates varying from 10 s to
thousands of cm-1.70-74 However, the rate of dephasing between
two sites will depend on the correlation function between site
energies.63 Indeed, the true dephasing strength for a crossconjugated unit will likely be quite weak, as the stochastic
fluctuations in site energy and intersite communication will
likely be strongly correlated within the small cross-conjugated
unit as the unit is of the same dimension as the individual solvent
molecules. The physical processes behind the estimates of
dephasing strengths are largely from energy fluctuations from
uncorrelated solvent motions between distal molecules in the
case of photon echo experiments75,76 or the fluctuating energy
gap between ground and excited states in resonance Raman
experiments.73 Contrasting these uncorrelated energies with the
largely correlated site energies in the small cross-conjugated
unit suggests these above estimates should be regarded as an
upper limit for dephasing strength.
For comparison, the level of dephasing required for erasing
the effect of interference on transport in benzene, a similarly
sized molecule to the cross-conjugated unit, is significantly
smaller; yet these interference effects persist, and have been
experimentally measured,34 and dominate the substitution
chemistry of phenyl compounds. Consequently, we believe that
the transport interference calculated in cross-conjugated molecules will survive electronic dephasing.
Molecular Dynamics
The transport through a molecule would ideally be constant
despite room temperature molecular vibrations and binding site
fluctuations. Experimental measurements on single molecule
conductance have primarily been made using Au electrodes.26,27,77-80 The Au surface is quite mobile, and the
distribution of conductance values through a sulfur-terminated
molecule can be quite large. This variation has been well studied
both experimentally and theoretically, with typical distributions
of (60% in conjugated molecules on Au surfaces.77,78,81-84 The
unique conductance characteristics of cross-conjugated mol-
Quantum Interference
ecules are a local effect and should be independent of variations
in electrode coupling. To study the interference stability,
molecular dynamics simulations have been used to give a
thermal geometric distribution.85 We focus our analysis on the
effect of internal fluctuations on the conductance of the
molecule. The molecular dynamics simulations were completed
using Tinker 4.0.86 A 1 ns trajectory was run to fully equilibrate
the system. For the next 100 ps, a snapshot was taken every 1
ps, and the geometry was parsed. Figure 8a and b shows the
transmission plots calculated in Hückel IV 3.0 for 100 geometries for the cross-conjugated molecule 9 and the linearly
conjugated molecule 14. The interference calculated in the crossconjugated molecules is stable to geometric fluctuations. The
variation in conductance is of the same order of magnitude in
all three molecules 5, 9, and 14.
The statistical conductance distribution is seen in Figure 8c
and d, which shows a Gaussian fit to a histogram of the transport
calculated through 100 molecular geometries. The histogram
bin width is set to 2*interquartile range*number of data
points-(1/3), allowing a direct comparison of the Gaussian fits.87,88
The conjugated molecule 14 has a 1.5× increase in conductance
when the voltage is increased from 10 mV to 2 V. The molecule
with saturated atoms 5 has a 10× increase in conductance, while
the cross-conjugated molecules 9 show a >90× increase in
conductance. Together, the dephasing calculations and the
molecular dynamics simulations provide strong evidence that
actually measuring the transport through cross-conjugated
molecules will show the effects of interference features.
Discussion
We have shown that quantum interference is quite common
in model systems. The unique transport behavior calculated in
model systems correlates well with the behavior calculated for
cross-conjugated molecules. From bond length analysis, we
calculate uncorrelated bond lengths and associated electron
delocalization across the cross-conjugated bond. This unique
feature of cross-conjugated molecules is borne out in the
interference position and in the unique transport behavior
calculated for these molecules. Geometric molecular distributions and dephasing are shown not to destroy this interference
feature, increasing the possibility of experimental realization.
Our methods used to calculate interference in transport do not
attempt a sophisticated treatment of electron correlation, because
all three are mean-field approaches (as is appropriate for
tunneling, coherent transport). The effect of electron-electron
correlation on charge transport and the role of poles in Green’s
function matrices have been studied by others.89-91 Understanding such quantum interference will allow the design of novel
molecular systems for device applications that require nonlinear
voltage dependence, including but not limited to molecular
current sensors and transistors.
Acknowledgment. This work was funded by NSF-Chemistry
(CHE-0719420, CHE-0414554, CHE-0718928), NSF-MRSEC
(DMR-0520513), ONR-Chemistry, NSF International Division,
the Purdue NCN, and the American Australian Foundation. We
would like to thank Atomistix for use of their software.
Supporting Information Available: Calculation details, a
comparison between transport codes for Figure 6, linear fit to
the IV data in Figure 6, parametric plots for the transmission
phase through the two-site model, details on the calculation of
transmission from the coupling, and complete ref 54. This
material is available free of charge via the Internet at http://
pubs.acs.org.
J. Phys. Chem. C, Vol. 112, No. 43, 2008 16997
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